An abstract Cauchy problem for higher order functional differential inclusions with infinite delay (Q2906302)

The authors consider the following Cauchy problem for the higher-order differential inclusion with infinite delay NEWLINE\[NEWLINE\begin{cases} u^{(N)}(t)+\sum_{i=0}^{N-1}A_iu^{(i)}(t)\in F(t,u(t),u_t),\quad t\in [0,T],\\ u^{(i)}(0)=u_i, \quad i=1,\dotsc,N-1,\\ u(s)=\phi(s), \quad s\in (-\infty,0], \end{cases}NEWLINE\]NEWLINE where \(N\geq 1\), \(A_i\), \(i=0,\dotsc,N-1,\) are linear operators in a Banach space \((X,\|\cdot\|)\), \(F\) is a multivalued map and \(u_t(s)=u(t+s)\), \(s\in (-\infty,0]\).NEWLINENEWLINEExistence results are obtained by using the concept of existence family, the notion of measure of non-compactness and a fixed point theorem for multivalued condensing maps. An application to some classes of partial differential equations is also presented.